q-Analogues of the Riemann zeta, the Dirichlet L-functions, and a crystal zeta function

Abstract

A q-analogue of the Riemann zeta function was studied in [Kaneko et al. 03] via a certain q-series of two variables. We introduce in a similar way a q-analogue of the Dirichlet L-functions and make a detailed study of them, including some issues concerning the classical limit of the q-analogue left open in [Kaneko et al. 03]. We also examine a "crystal" limit (i.e. q->0) behavior of the q-analogue. The q-trajectories of the trivial and essential zeros of the Riemann zeta function are investigated numerically when q moves in (0,1]. Moreover, conjectures for the crystal limit behavior of zeros of the q-analogue are given.

24 pages, 15 figures

References in corpus (1)

• q-Multiple Zeta Functions and q-Multiple Polylogarithms

Cited by in corpus (7)

• q-Multiple Zeta Functions and q-Multiple Polylogarithms
• A twisted spectral triple for quantum SU(2)
• Integral representations of q-analogues of the Hurwitz zeta function
• Evaluations of multiple Dirichlet $L$-values via symmetric functions
• Renormalization of Multiple $q$-Zeta Values
• q-Analogues of the Barnes multiple zeta functions
• Nonstandard Mathematics and New Zeta and L-Functions